where \(M_p^2=8\pi G\)  is the reduced Planck mass, \(L\)  is the size of a system (thus \(L^{2}\)  is essentially the area of the boundary surface). This relation between the energy density in a bulk and the area of its space-time boundary surface is rooted in the holographic principle [1]}. The argument that leads to the above relation is the following. The zero-point energy diverges quartically and thus the energy density scales as \(\rho _\Lambda \sim \Lambda ^4\)  if we simply cut-off the divergence at the UV scale \(\Lambda \)  Noted that there are some works [2]}[3]}[4]} that challenge this perspective.. However, this simple scaling is violated when the total energy in the system with size \(L\) , i.e. \(L^3\Lambda ^4\) , approaches to the mass of the black hole in the same size \(LM_p^2\) . In fact, the total energy must be bounded by the mass of the black hole from above, as the effective quantum field theory breaks down at the Schwarzschild radius scale. This energy bound is stronger than the Bekenstein entropy bound [5]} which inspired the proposal of the holographic principle [1]}. Nowadays it is widely believed that the holographic principle is one of the most important cornerstones of quantum gravity.
